mathsurgery - GCSE GM8

EPCHS GCSE Scheme of Work for first teaching 2007-2009

Unit S4 Angles, bearings & construction.

Web Links

General websites about geometric constructions

A range of video clips (now downloaded to the mathsbase folder) and explanations at:
http://regentsprep.org/Regents/math/math-topic.cfm?TopicCode=construc

For a list of standard constructions with step by step guides use:
http://whistleralley.com/construction/reference.htm

A historical overview and links to more sophisticated results at:
http://mathworld.wolfram.com/GeometricConstruction.html

For a constriction toolkit go to:
http://www.suffolkmaths.co.uk/pages/Top%20Activities/Teaching%20Techniques/Shape%20and%20Space/Loci/Resources/Construction%20Kit/Construction%20Kit.html

Another useful tool for drawing combined loci at:
http://www.ngfl-cymru.org.uk/vtc/ngfl/maths/echalk/lociWeb/
… and bearings…
http://www.ngfl-cymru.org.uk/vtc/ngfl/maths/echalk/bearings/

For specific references to learning objectives use this table:

level	grade	Learning objective	Web link
10a	A*	use congruence and other angle properties to rigorously *prove* standard constructions work (e.g. perpendicular bisector construction generates the vertices of a rhombus, and congruence conditions of triangles to prove that that the diagonals *must* generate 4 congruent right angled triangles) experience constructing triangles given ASS (where in the usual notation we are given angle A and sides a and b) using a circle to construct the 3rd vertex ensure examples include all four cases for the 2nd side: 1.a = b sin A Þ the result is of the form RHS since a right angle results as the indeterminate ray is a tangent to the circle 2.a < b sin A Þ no triangle is possible 3.a > b Þ the result is a unique triangle since A lies inside the circle radius a centre C 4.b sin A < a < b Þ there are two distinct triangles satisfying these conditions understand that SAA guarantees congruence since the data is sufficient to determine all the angles (angle sum of triangle) and hence SAA is equivalent to ASA understand and experience by doing several constructions that ASA and SAS guarantee congruence understand that SSS guarantees congruent triangles but AAA only guarantees similar triangles
10b
10c
9a	A
9b
9c
8a	B	determine when the data for a triangle are insufficient to make a unique triangle use symmetry and shape properties to discuss why standard constructions work (e.g. perpendicular bisector works because the construction generates the vertices of a rhombus, and we can use symmetry and shape properties to convince ourselves that the diagonals *ought to* generate 4 congruent right angled triangles) construct circumcircles and inscribed circles to triangles determine more complex loci including those determined by fixed distances from line segments or polygons and the parabola determined by equidistance from a focus and directrix constructing regular polygons animation contextual problem involving a number of loci conditions [animation] **
8b
8c
7a	C	understand the link between locus and construction determine when, given SSS or AAA, a triangle is impossible and explain how this relates to any attempted construction maths4real video on loci associated maths4real pdf worksheet challenge: construct this shape [web page] **
7b		determine the locus of points which are equidistant from a pair of points (perpendicular bisector) determine the locus of points which are equidistant from a pair lines (angle bisector)
7c		construct more complicated arrangements of shapes such as nets of prisms and pyramids
6a	D	determine the locus of points which are equidistant from a point (circle definition, consider sphere as 3D analogue) constructing an equilateral triangle [animation] constructing a parallel to a line through a point [animation] constructing a perpendicular line through a point [animation]
6b		use a straight edge and compasses to construct a right angled triangle given right angle, hypotenuse and side, constructing the right angle as below and finding the third side by intersection of line and arc
6c		use a straight edge and compasses to find a perpendicular from a point to a line and to find a perpendicular from a point on a line
5a	E	use a straight edge and compasses to find an ‘angle bisector’ and I know what this means constructing a perpendicular bisector [animation] constructing a perpendicular bisector to a line segment between two point [animation] bisect an angle [video] bisect a line segment [video]
5b		use a straight edge and compasses to find a ‘perpendicular bisector’ to a ‘line segment’ and know what these words mean
5c		draw a triangle using a ruler and compasses given the three lengths
4a	F	use standard ways of labelling lines, angles and shapes to label drawings and interpret instructions: e.g. draw line PQ with length 5.6cm, BÂC = 60° copying a line segment [video] copying an angle [video] holding and using a pair of compasses [web page] drawing circles [series of Cabri files]
4b		draw an angle to within 1° construct triangles given two lengths and the angle between them construct triangles given two angles and the length between them
4c		use a pair of compasses to draw circles and arcs of a given radius and make these accurate to the nearest mm
3a	G	draw an angle correct to within 5° perhaps using a simplified angle measurer (protractor)
3b		draw lines correct to the nearest mm, even when given a length such as 5.6cm or 5cm 6mm
3c		draw an angle that is larger or smaller than a given angle (less than 180°), or between two others (both less than 180°)

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GCSE GM8